Journal d’Analyse Mathematique

, Volume 55, Issue 1, pp 117–171

Moduli spaces of quadratic differentials

  • William A. Veech
Article

Abstract

The cotangent bundle ofJ (g, n) is a union of complex analytic subvarieties, V(π), the level sets of the function “singularity pattern” of quadratic differentials. Each V(π) is endowed with a natural affine complex structure and volume element. The latter contracts to a real analytic volume element, Μπ, on the unit hypersurface, V1(π), for the Teichmüller metric. Μπ is invariant under the pure mapping class group, γ(g, n), and a certain class of functions is proved to be Lpπ), 0 <p < 1, over the moduli space V1(π)/γ (g, n). In particular, Μπ(V1(π)/γ(g, n)) < ∞, a statement which generalizes a theorem by H. Masur.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B83]
    L. Bers,Finite Dimensional Teichmüller spaces and generalizations, Proc. Symp. Pure Math.39 (1983), 115–156.Google Scholar
  2. [Bi74]
    J. S. Birman,Braids, Links, and Mapping Class Groups, Annals of Mathematics Studies, Number 82, Princeton University Press, Princeton, N.J., 1974.Google Scholar
  3. [FLP79]
    A. Fathi, F. Laudenbach, V. Poènaru,et al.,Travaux de Thurston sur les surfaces, Asterisque66–67 (1979).Google Scholar
  4. [LV73]
    O. Lehto and K. I. Virtanen,Quasiconformal Mappings in the Plane, Springer-Verlag, New York, 1973.MATHGoogle Scholar
  5. [M43]
    A. G. Maier,Trajectories on closed orientable surfaces, Math. Sb.12(54) (1943), 71–84 (Russian).Google Scholar
  6. [M82]
    H. Masur,Interval exchange transformation and measured foliations, Ann. of Math.115 (1982), 169–200.CrossRefMathSciNetGoogle Scholar
  7. [M84]
    H. Masur,Ergodic actions of the mapping class group, preprint.Google Scholar
  8. [Mo66]
    C. Moore,Ergodicity of flows on homogeneous spaces, Am. J. Math.88 (1966), 154–178.MATHCrossRefGoogle Scholar
  9. [R79]
    G. Rauzy,Echanges d’intervalles et transformations induites, Acta Arith.34 (1979), 315–328.MATHMathSciNetGoogle Scholar
  10. [S84]
    K. Strebel,Quadratic Differentials, Springer-Verlag, Berlin, 1984.MATHGoogle Scholar
  11. [Th76]
    W. Thurston,On the dynamics of diffeomorphisms of surfaces, preprint.Google Scholar
  12. [Th83]
    W. Thurston,The geometry and topology of 3-manifolds, preprint.Google Scholar
  13. [V82]
    W. A. Veech,Gauss measures for transformations on the space of interval exchange maps, Ann. of Math.115 (1982), 201–242.CrossRefMathSciNetGoogle Scholar
  14. [V84]
    W. A. Veech,Dynamical systems on analytic manifolds of quadratic differentials. I. F-strctures. II. Analytic manifolds of quadratic differentials, preprints, 1984.Google Scholar
  15. [V86]
    W. A. Veech,The Teichmüller geodesic flow, Ann. of Math.115 (1986), 441–530.CrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 1990

Authors and Affiliations

  • William A. Veech
    • 1
  1. 1.Department of MathematicsRice UniversityHoustonUSA

Personalised recommendations